is too large, and the unscaled function is required, there is a risk of overflow and so no computation is performed. In all the above cases, a warning is given by the function.

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Amos D E (1986) Algorithm 644: A portable package for Bessel functions of a complex argument and non-negative order ACM Trans. Math. Software 12 265–273

Parameters

Compulsory Input Parameters

1: $deriv$ – string (length ≥ 1)

Specifies whether the function or its derivative is required.

$deriv ='F'$: $Ai (z)$ is returned.
$deriv ='D'$: ${Ai}^{'} (z)$ is returned.

Constraint:

deriv ='F'

'D'

2: $z$ – complex scalar

The argument

z

of the function.

3: $scal$ – string (length ≥ 1)

The scaling option.

$scal ='U'$: The result is returned unscaled.
$scal ='S'$: The result is returned scaled by the factor $e^{2 z \sqrt{z} / 3}$ .

Constraint:

scal ='U'

'S'

Optional Input Parameters

None.

Output Parameters

1: $ai$ – complex scalar

The required function or derivative value.

2: $nz$ – int64int32nag_int scalar

Indicates whether or not ai is set to zero due to underflow. This can only occur when

scal ='U'

$nz = 0$: ai is not set to zero.
$nz = 1$: ai is set to zero.

3: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,	$deriv \neq'F'$ or $'D'$ .
or	$scal \neq'U'$ or $'S'$ .

$ifail = 2$: No computation has been performed due to the likelihood of overflow, because $Re (w)$ is too large, where $w = 2 z \sqrt{z} / 3$ – how large depends on z and the overflow threshold of the machine. This error exit can only occur when $scal ='U'$ .

W $ifail = 3$: The computation has been performed, but the errors due to argument reduction in elementary functions make it likely that the result returned by nag_specfun_airy_ai_complex (s17dg) is accurate to less than half of machine precision. This error exit may occur if $abs (z)$ is greater than a machine-dependent threshold value.

$ifail = 4$: No computation has been performed because the errors due to argument reduction in elementary functions mean that all precision in the result returned by nag_specfun_airy_ai_complex (s17dg) would be lost. This error exit may occur if $abs (z)$ is greater than a machine-dependent threshold value.

$ifail = 5$: No result is returned because the algorithm termination condition has not been met. This may occur because the arguments supplied to nag_specfun_airy_ai_complex (s17dg) would have caused overflow or underflow.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

All constants in nag_specfun_airy_ai_complex (s17dg) are given to approximately

18

digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used

t

, then clearly the maximum number of correct digits in the results obtained is limited by

p = \min (t, 18)

. Because of errors in argument reduction when computing elementary functions inside nag_specfun_airy_ai_complex (s17dg), the actual number of correct digits is limited, in general, by

p - s

, where

s \approx \max (1, |\log_{10} |z||)

represents the number of digits lost due to the argument reduction. Thus the larger the value of

|z|

, the less the precision in the result.

Empirical tests with modest values of

z

, checking relations between Airy functions

Ai (z)

{Ai}^{'} (z)

Bi (z)

and

{Bi}^{'} (z)

, have shown errors limited to the least significant

3

–

4

digits of precision.

Further Comments

Note that if the function is required to operate on a real argument only, then it may be much cheaper to call nag_specfun_airy_ai_real (s17ag) or nag_specfun_airy_ai_deriv (s17aj).

Example

This example prints a caption and then proceeds to read sets of data from the input data stream. The first datum is a value for the argument deriv, the second is a complex value for the argument, z, and the third is a character value to set the argument scal. The program calls the function and prints the results. The process is repeated until the end of the input data stream is encountered.

Open in the MATLAB editor: s17dg_example

function s17dg_example


fprintf('s17dg example results\n\n');

z  = [0.3 + 0.4i;  0.2 + 0i;  1.1 - 6.6i;  1.1 - 6.6i; -1 + 0i];
deriv = {'F';      'F';       'F';         'F';         'D'};
scal  = {'U';      'U';       'U';         'S';         'U'};

fprintf('deriv        z        scaled?         Ai(z)       nz\n');
for i=1:numel(z)

  [cy, nz, ifail] = s17dg(deriv{i}, complex(z(i)), scal{i});

  fprintf('  %s  %7.3f%+7.3fi', deriv{i}, real(z(i)), imag(z(i)));
  if scal{i} == 'U'
     fprintf('  unscaled');
  else
     fprintf('    scaled');
  end
  fprintf(' %7.3f%+8.3fi %3d\n', real(cy), imag(cy), nz);
end

s17dg example results

deriv        z        scaled?         Ai(z)       nz
  F    0.300 +0.400i  unscaled   0.272  -0.100i   0
  F    0.200 +0.000i  unscaled   0.304  +0.000i   0
  F    1.100 -6.600i  unscaled -43.663 -47.903i   0
  F    1.100 -6.600i    scaled   0.165  +0.060i   0
  D   -1.000 +0.000i  unscaled  -0.010  +0.000i   0

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox